The Holey Monster (with 934 faces) - Numberphile

Episode Overview

• The video explores the world of complex polyhedra, inspired by B.M. Stewart's hand-illustrated book, "Adventures Among the Toroids."
• It explains how different families of geometric shapes, from Platonic to Archimedean and Johnson solids, are defined by progressively relaxing rules of symmetry and composition.
• The discussion introduces "Stewart Toroids," a special class of non-convex polyhedra characterized by having holes or "tunnels" while still adhering to specific geometric criteria.
• The episode showcases several visually intricate shapes, including the record-breaking "Holey Monster," which has an exceptionally high number of holes (genus).

Key Concepts

• Platonic Solids: The five convex polyhedra with identical, regular polygonal faces where all vertices are the same (e.g., a cube). Their key properties include regular faces and interchangeable faces, edges, and corners.
• Archimedean Solids: A group of 13 polyhedra with regular polygonal faces of two or more types. They are created by relaxing the "interchangeable faces" property of Platonic solids while keeping vertices identical (e.g., cuboctahedron, truncated cube).
• Johnson Solids: A set of 92 convex polyhedra made from regular polygonal faces. They are not as symmetrical as Platonic or Archimedean solids because the arrangement of faces around each vertex is not necessarily the same.
• Stewart Toroids: A specific class of non-convex polyhedra defined by four criteria: 1) Regular polygonal faces, 2) Quasi-convexity (its convex hull doesn't create new edges), 3) A-planarity (no adjacent faces are coplanar), and 4) Being "tunnelled" (having at least one hole).
• Genus: A topological property that describes the number of "holes" a shape has. A sphere or cube has a genus of 0, while a donut-shaped torus has a genus of 1.

Quotes

• At 00:20 - "This whole book is written by hand in calligraphy, all the pictures are hand-drawn... it's just this absolutely magnificent thing and such a labor of love." - The speaker highlights the unique and artistic nature of the book "Adventures Among the Toroids" that inspired the discussion.
• At 01:55 - "A lot of the theory of polyhedra develops by just saying, 'Okay, we'll let's just look at that list of properties and then slightly relax some or tweak some, and you get a whole load of interesting other shapes.'" - Explaining the methodology for discovering new classes of polyhedra by modifying the strict rules that define simpler shapes like the cube.
• At 16:01 - "He called this the Holey Monster, H-O-L-E-Y." - Introducing the nickname for a Stewart Toroid with a record-breaking genus (number of holes) of 46.

Takeaways

• The classification of geometric shapes is a systematic process of defining properties and then relaxing them to discover new families, from the highly symmetric Platonic solids to the more varied Johnson solids.
• A polyhedron can have holes and still be defined by rigorous mathematical rules, as seen in the "Stewart Toroids."
• The "genus" of a shape, or its number of holes, is a key characteristic, leading mathematicians to create complex structures like the "Holey Monster" in the pursuit of maximizing this property within a set of rules.
• Convexity is a fundamental property in geometry, and its opposite (non-convexity) opens the door to studying shapes with tunnels and intricate internal structures.